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Re: Problem with LaplaceTransform and InverseLaplaceTransform
*To*: mathgroup at smc.vnet.net
*Subject*: [mg67541] Re: Problem with LaplaceTransform and InverseLaplaceTransform
*From*: ab_def at prontomail.com
*Date*: Fri, 30 Jun 2006 04:14:26 -0400 (EDT)
*References*: <e7td5h$3kh$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
aXi wrote:
> Can someone point me to site that would help me in using Mathematica
> 5.2for purpose of calculations related to Systhems theory. I'm having
> trouble with several topics... for example doing
> InverseLaplaceTransform of function:
>
> (1/s) * (TanH[pi*s/2])
>
> Thanks in advance!
If you want the inverse Laplace transform of Tanh[Pi*s/2]/s, it can be
found as follows. First take the transform of Tanh[Pi*s/2]:
In[1]:= fs = Tanh[Pi*s/2]/s;
In[2]:= InverseLaplaceTransform[s*fs, s, t]
Out[2]= DiracDelta[t] + 2*Sum[(-1)^K$40*DiracDelta[(-K$40)*Pi + t],
{K$40, 1, Infinity}]
Then integrate the result:
In[3]:= Integrate[DiracDelta[t], t] +
2*Sum[Integrate[(-1)^k*DiracDelta[t - Pi*k], t], {k, Infinity}]
Out[3]= UnitStep[t] + (-1 + (-1)^Floor[t/Pi])*UnitStep[-1 + t/Pi]
Strictly speaking, we need to evaluate Integrate[f[tau], {tau, 0-, t}],
'including' the zero in the integration range.
It is easy to see that the result is equal to (-1)^Floor[t/Pi] or
Sign[Sin[t]]. Here's a check:
In[4]:= LaplaceTransform[Sign[Sin[t]], t, s] - fs // Simplify
Out[4]= 0
Maxim Rytin
m.r at inbox.ru
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